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Exponents and Polynomials
Chapter 5
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5.1
Adding and Subtracting Polynomials
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5.1 Adding and Subtracting Polynomials
Objectives
1. Review combining like terms.2. Know the vocabulary for polynomials.3. Evaluate polynomials.4. Add polynomials.5. Subtract polynomials.6. Add and subtract polynomials with more than
one variable.
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5.1 Adding and Subtracting Polynomials
Review Combining Like Terms
(a) – 3 x + 7 x 5 5
= (– 3 + 7) x 5
= 4 x 5
= (2 + 5 – 1) n 3
= 6 n 3
(b) 2 n + 5 n – n 3 3 3
Distributive property
Example 1 Simplify each expression by adding like terms.
Distributive property
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5.1 Adding and Subtracting Polynomials
Review Combining Like Terms
(c) 4 g h – 9 g h + 2 g h 7 7 7 4 4 4
= (4 – 9 + 2) g h 7 4
= – 3 g h 7 4
Example 1 (concluded)Simplify each expression by adding like terms.
Distributive property
Example 2 Simplify each polynomial if possible. Then give the degree and tell whether the polynomial is a monomial, a binomial, a trinomial, or none of these.
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5.1 Adding and Subtracting Polynomials
Knowing the Vocabulary for Polynomials
(a) 2 t + 7 4 The polynomial cannot be simplified.The degree is 4.The polynomial is a binomial.
The polynomial can be simplified.
The degree is 2.The simplified polynomial is
a monomial.
(b) 3 e + 5 e – 9 e2 2 2
= – e 2
Two terms.
One term.
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5.1 Adding and Subtracting Polynomials
Evaluating a Polynomial
Substitute h = –3.
Example (a) Find the value of 5h4 – 3h2 + 4h + 7 when h = – 3.
Apply the exponents.
Multiply.
Add and subtract.
4 25 3 4 7h h h
4 23 3 4 35 3 7
5 81 3 9 4 3 7
405 27 12 7
373
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5.1 Adding and Subtracting Polynomials
Evaluating a Polynomial
Substitute h = 2.
Example (concluded) (b) Find the value of 5h4 – 3h2 + 4h + 7 when h = 2.
Apply the exponents.
Multiply.
Add and subtract.
4 25 3 4 7h h h
4 22 25 3 4 72
5 16 3 4 4 2 7
80 12 8 7
83
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5.1 Adding and Subtracting Polynomials
Evaluating a Polynomial
CAUTIONUse parentheses around the numbers that are substitutedfor the variable in Example 3, particularly when substitutinga negative number for a variable that is raised to a power. Otherwise, a sign error may result.
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5.1 Adding and Subtracting Polynomials
Adding Polynomials
Adding PolynomialsTo add two polynomials, add like terms.
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5.1 Adding and Subtracting Polynomials
Adding Polynomials
Example 4
8y – 7y – y + 33 2
6y + 2y – 4y + 13 2
+ 4– 5y 14y 3 2 – 5y
Write like terms in columns.
Now add, column by column.
3 2 3 2(a) Add 8 7 3 and 6 2 4 1.y y y y y y
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5.1 Adding and Subtracting Polynomials
Adding Polynomials
+ 12 – 7n 7n 3 2+ n
Write like terms in columns.
Now add, column by column.
n – 9n + 122
7n 3 + 2n
Example 4 (concluded)3 2(b) Add 7 2 and 9 12.n n n n
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5.1 Adding and Subtracting Polynomials
Adding Polynomials
– 3n 4 – 15n 3 + 6=
Example 54 3 4 3(a) Add 2 7 4 and 5 8 10.n n n n
( 2n – 7n – 4 ) + ( – 5n – 8n + 10 )4 3 4 3
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5.1 Adding and Subtracting Polynomials
Adding Polynomials
7p 3 – 4p – 5=
3
Example 5 (concluded)3(b) Add 7 9 4 and 5 1.p p p
( 7p – 9p – 4 ) + ( 5p – 1 )
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5.1 Adding and Subtracting Polynomials
Subtracting Polynomials
Subtracting PolynomialsTo subtract two polynomials, change all the signs of the second polynomial and add the result to the first polynomial.
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5.1 Adding and Subtracting Polynomials
Subtracting Polynomials
( 3x – 5 ) – ( 6x – 4 )
Change the signs in the second polynomial and add.
– 3x = – 1
Example 6 (a) Perform the subtraction 3 5 6 4 .x x
= ( 3x – 5 ) + ( – 6x + 4 )
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5.1 Adding and Subtracting Polynomials
Subtracting Polynomials
Change the signs in the second polynomial and add.
=
( 7y + 8 ) – ( y + 4y – 2 )3 3 2
6y 3 – 4y 2 + 10
Write the problem.
Example 6 (concluded)3 2 3(b) Subtract 4 2 from 7 8.y y y
= ( 7y + 8 ) + ( – y – 4y + 2 )3 3 2
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5.1 Adding and Subtracting Polynomials
Subtracting Polynomials
Arrange like terms in columns..
4g + 6g – g – 54 3
– 6g + 2g – 4g + 34 3 6g – 2g + 4g – 34 3
4g + 6g – g – 54 3
Change all signs in the second row, then add.
+ 4g 3 + 3g – 810g 4
Example 7
4 3 4 3
(b) Subtract by columns:
4g 6 5 6g 2 4 3 .g g g g
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5.1 Adding and Subtracting Polynomials
Subtracting Polynomials
Example 8 Perform the indicated operations to simplify the expression.
Rewrite, changing the subtraction to adding the opposite.
Combine like terms.
5= – 8m 2+ 15m
( 2 – 3m + 5m ) + ( – 6 – 4m + 7m ) + ( 9 – m + 3m )2 2 2 =
( 2 – 3m + 5m ) – ( 6 + 4m – 7m ) + ( 9 – m + 3m ).2 2 2
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5.1 Adding and Subtracting Polynomials
Adding and Subtracting Multivariable Polynomials
Example 9
(a) Add or subtract as indicated.
= 9c + 7cd – 3d
( 6c – 2cd + d ) + ( 3c + 9cd – 4d )
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5.1 Adding and Subtracting Polynomials
Adding and Subtracting Multivariable Polynomials
Example 9 (concluded)
(b) Add or subtract as indicated.
– ab
( 2a b – 4ab + b ) – ( 5a b – 3ab + 7b )2 2 2 2
2 2 2 2
= – 3a b2 – 6b 2
= 2a b – 4ab + b – 5a b + 3ab – 7b